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If you think of an ordinary integral $\int _ a ^ b g ( x ) \, \mathrm d x$ as an area (or a signed area in case $g$ might take negative values), then you can think of a double integral $\int _ R ... • 4,714 15 votes Accepted Show$\int_0^{\pi/2}\int_0^1 e^{t+t^{\tan\theta}}dtd\theta=\frac{\pi}{4}(e^2-1)$First begin with the substitution$s=\tan\theta\implies \frac{1}{1+s^2}ds=d\theta, giving us \begin{align*}I&:=\int_{0}^{\frac{\pi}{2}}\int_{0}^{1}e^{t+t^{\tan\theta}}\, dt\, d\theta \\ &\,=\... • 5,099 12 votes Interpreting a notation in calculus of variations (differentiating with respect to a derivative) Consider a functionalJ[y]$defined by: $$J[y] = \int_a^b F(x, y, y') dx \tag{1}$$ I think it will be helpful to remind you right away that your notation implies that you have already agreed to ... • 556 10 votes A tricky system of non-linear multivariate equations Add the first equation to twice the second and to the third and express the result as $$(x+a)^2+(y+b)^2=(z+c)^2$$ Draw the following triangles to illustrate this equation as well as the equations $$... • 21.8k 9 votes Inequality \sum_{k=1}^{n} \frac{\log(a_k)}{1+a_{k}^{2}} \leqslant 0 We have$$ 2 \sum_{k=1}^{n} \frac{\log(a_k)}{1+a_{k}^{2}} = \sum_{k=1}^{n} \left( \frac{2\log(a_k)}{1+a_{k}^{2}} - \log(a_k)\right) \\ = \sum_{k=1}^{n}\frac{(1-a_k^2)\log(a_k)}{1+a_{k}^{2}} \le 0 $$... • 113k 9 votes Accepted Do double integrals always give a volume? The topic of double integrals$$\iint_D f(x,y) \ dxdy$$affirms that from the geometric point of view it can be interpreted as the volume of the solid obtained by extending the plane region D until ... • 7,649 8 votes Accepted Find limit using generalized binomial theorem. You have exactly the right idea! As a life-pro-tip, one common way to simplify estimates like these is to try to get all of our norms to look similar. Here we have a |x+y| in the numerator (which ... • 38.2k 8 votes Accepted How to evaluate \displaystyle \int_{-\pi/2}^{\pi/2} f(x) dx where f(x)=\cos(x)+\sin(f(x)) \def\J{\operatorname J} \def\H{\operatorname H} Apply the Kepler equation Fourier series, so Bessel J appears$$\text{Area}=\int_{-\frac\pi2}^\frac\pi2\cos(x)+\sum_{n=1}^\infty\frac{2\sin(n\cos(x))}... • 12.1k 8 votes Accepted Matrix Derivative of$X^3$Since matrix multiplication is not commutative, you have to be more careful and more explicit. Compute the derivative in the direction of a tangent vector (i.e., matrix)$H: \begin{align*} Df_A(H) &... • 115k 8 votes Accepted Integrate\int_0^1\left(\int_0^\pi \frac{u}{\sqrt{1+u^2-2u \cos\phi}} d\phi\right)du$With$K$as the Complete Elliptic Integral of the First Kind with Paramater$k, $$K:=K(k)=\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2\sin^2t}}dt$$ Also using Landen's Transformation: $$K\left(\frac{2\sqrt{x}}{... • 3,230 8 votes Accepted How to calculate \int _0^1 \int _0^1\left(\frac{1}{1-xy} \ln (1-x)\ln (1-y)\right) \,dxdy$$ \begin{align} &\hspace{5mm}\int _0^1 \int _0^1 \frac{\ln (1-x)\ln (1-y) }{1-xy} \,dxdy \\&=\int_0^1 \ln(1-x)\frac{Li_2(\frac x{1-x})}{x}dx =\int_0^1 {Li_2(\frac x{1-x})}d[Li_2(1)-Li_2(x)]\\ ... • 97.7k 7 votes Accepted A clean way to identify\nabla(|f|^2)$with$2f\overline{f'}$for a holomorphic$f$. If you identify$\nabla W$with$\frac{\partial}{\partial x}W + i \frac{\partial}{\partial y}W then you can use the Wirtinger derivatives {\begin{aligned}{\frac {\partial }{\partial z}}&={\... • 113k 7 votes vector-valued function visualization You just draw the vector (x^2-y^2,2xy) at position (x,y): • 18.1k 6 votes Accepted Asymptotic behavior of the inverse Fourier transform of \frac{1}{|k|^2 + 1}f(x)=\int_{-\infty}^\infty...\int_{-\infty}^\infty\frac {e^{ik_1x_1+ik_2x_2+...+ik_nx_n}}{1+k_1^2+...+k_n^2}dk_1...dk_n=\int_{-\infty}^\infty...\int_{-\infty}^\infty e^{ik_1x_1+ik_2x_2+...+... • 15.7k 6 votes How to check if the limit exists in multivariable calculus You can use the estimate $$\bigg| y e^{-\frac{1}{\sqrt{x^2 + y^2}}}\bigg| \le \sqrt{x^2 + y^2} \cdot e^{-\frac{1}{\sqrt{x^2 + y^2}}},$$ switch to polar coordinates and compute the limit $$\lim_{r \... • 3,656 6 votes Accepted How to check if the limit exists in multivariable calculus You idea is perfectly fine and I think that hints or suggestions given are a little bit confusing. Indeed squezee theorem and polar coordinates are really not necessary in this case. We can simply ... • 155k 6 votes How do I evaluate \int_{0}^{\infty}{e^{-x}\frac{\sin^2{x}}{x}dx} = \frac{\ln{5}}{4} using iterated integrals? Amusingly, I solved this as part of a homework assignment last semester. The hint that I was given would probably prove helpful though: verify and use the fact that$$ \int_0^1 e^{-y} \sin(2xy) \, \... • 44k 6 votes Accepted The directional derivative equals dot product of gradient and a unit vector. But what if the function is not totally differentiable? I think most of what needs to be said is in the comments, but in an attempt to set things out clearly: Iff\colon \Omega\to \mathbb R$is a function defined on an open set$\Omega$of$\mathbb R^n$(... • 4,568 6 votes Root test application question (usefulness of root test for the series$\sum 1/n^n$) Ratio test does work: Let$a_n=1/n^n$and it follows $$\left|\frac{a_{n+1}}{a_n}\right|=\frac{n^n}{(n+1)^{n+1}}\leq\frac{n^{n+1}}{(n+1)^{n+1}}=\left(1-\frac1{n+1}\right)^{n+1}\to\frac1e\quad\text{as }... • 2,587 6 votes Using infinitesimals in multivariable calculus There are pitfalls with using notation that is too abbreviated when working in multivariate calculus, whether or not one uses infinitesimals. Partial derivatives are treated carefully in Keisler's ... • 42.6k 6 votes Minimizing \sum\limits_{cyc}\frac{\sqrt{5a+8bc}}{8a+5bc} with \sum\limits_{cyc}ab=1 Some thoughts. Let$$x := 5a + 8bc, \quad y := 5b + 8ca, \quad z := 5c + 8ab,u := 8a + 5bc, \quad v := 8b + 5ca, \quad w := 8c + 5ab.$$Let$$A := \frac{2\sqrt{2} - \sqrt{5}}{3}, \quad B := \frac{... • 37.8k 6 votes Calculating Volume of "fat sphere" defined by$x^8+y^8+z^8 = 64$The technique is due to Dirichlet. It is spelled out fairly completely in the first edition of Whittaker and Watson, but reduced to a homework exercise by the fifth. Delighted to find that the 1902 ... • 140k 6 votes Accepted Finding the Inverse of a Function on the Sphere Your transformation can be written under a matrix form : $$\pmatrix{x'\\y'\\z'}=\pmatrix{\cos(z) & \sin(z) & 0\\ \sin(z) & -\cos(z) & 0\\0&0&1}\pmatrix{x\\y\\z}$$ Its inverse ... • 82k 6 votes Evaluating a double integral involving a biquadratic with variable coefficients Integrate w.r.t.$u$first; the denominator can be rearranged as $$\left(1035 t^4 + 2070 t^2 + 1035\right) u^2 + \left(729 t^4 + 594 t^2 + 121\right) = 1035 \left(t^2+1\right)^2 u^2 + \left(27t^2+11\... • 19.6k 6 votes Accepted How to integrate \int_{0}^{1} \int_{0}^{1} \ln\left(\frac{1}{\sinh^2(x) + \cosh^2(y)}\right) \,dx\,dy$$ \begin{split} J &= \int \frac{\ln(y(x^2+1)+x(y^2+1))}{xy}\, dx\\ & = \frac{1}{y} \int \frac{\ln(y(x^2+1)+x(y^2+1))}{x}\, dx \\ & = \frac{1}{y} \int \frac{\ln(x+y)+\ln(x+\frac{1}{y})+\ln(... • 108 6 votes Accepted How to integrate$\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dx\,dy$Hint: Per symmetry $$\int_{0}^{1} \int_{0}^{1} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) dxdy= 2\int_{0}^{1} \int_{0}^{y} \tanh^{-1}\left(\frac{x}{y} + \frac{y}{x}\right) \,dxdy$$ Then, ... • 97.7k 6 votes Accepted Derivative of$Ax x^\top A$with respect to$x$Let's look at pertubations $$f(x+v) = A(x+v)(x+v)^TA = Axx^TA + Axv^TA + Avx^TA + Avv^TA$$ The derivate is often defined as the unique linear function such that:$$f(x+v) = f(x) + D_{f;x}(v) + \... • 1,041 5 votes Accepted Is$\frac{\partial}{\partial x}(x-y)$ill-defined? The only way in which$\frac{\partial}{\partial x}(x-y)$is ill-defined, is the same way in which the function$x-y$is ill-defined. That is, the very instant we define the function$x-y\$, its ...
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